244 research outputs found
Message passing for the coloring problem: Gallager meets Alon and Kahale
Message passing algorithms are popular in many combinatorial optimization
problems. For example, experimental results show that {\em survey propagation}
(a certain message passing algorithm) is effective in finding proper
-colorings of random graphs in the near-threshold regime. In 1962 Gallager
introduced the concept of Low Density Parity Check (LDPC) codes, and suggested
a simple decoding algorithm based on message passing. In 1994 Alon and Kahale
exhibited a coloring algorithm and proved its usefulness for finding a
-coloring of graphs drawn from a certain planted-solution distribution over
-colorable graphs. In this work we show an interpretation of Alon and
Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus
showing a connection between the two problems - coloring and decoding. This
also provides a rigorous evidence for the usefulness of the message passing
paradigm for the graph coloring problem. Our techniques can be applied to
several other combinatorial optimization problems and networking-related
issues.Comment: 11 page
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest
possible colors. We present a randomized polynomial time algorithm that colors
a 3-colorable graph on vertices with min O(Delta^{1/3} log^{1/2} Delta log
n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any
vertex. Besides giving the best known approximation ratio in terms of n, this
marks the first non-trivial approximation result as a function of the maximum
degree Delta. This result can be generalized to k-colorable graphs to obtain a
coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)}
log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and
Williamson who used an algorithm for semidefinite optimization problems, which
generalize linear programs, to obtain improved approximations for the MAX CUT
and MAX 2-SAT problems. An intriguing outcome of our work is a duality
relationship established between the value of the optimum solution to our
semidefinite program and the Lovasz theta-function. We show lower bounds on the
gap between the optimum solution of our semidefinite program and the actual
chromatic number; by duality this also demonstrates interesting new facts about
the theta-function
Approximation Algorithms for Partially Colorable Graphs
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise.
We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
On the random satisfiable process
In this work we suggest a new model for generating random satisfiable k-CNF
formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k}
possible clauses over the variables x_1, ..., x_n, and starting from the empty
formula, go over the clauses one by one, including each new clause as you go
along if after its addition the formula remains satisfiable. We study the
evolution of this process, namely the distribution over formulas obtained after
scanning through the first m clauses (in the random permutation's order).
Random processes with conditioning on a certain property being respected are
widely studied in the context of graph properties. This study was pioneered by
Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and
also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite
graphs. Since then many other graph properties were studied such as planarity
and H-freeness. Thus our model is a natural extension of this approach to the
satisfiability setting.
Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently
large constant, we are able to characterize the structure of the solution space
of a typical formula in this distribution. Specifically, we show that typically
all satisfying assignments are essentially clustered in one cluster, and all
but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying
assignments. We also describe a polynomial time algorithm that finds with high
probability a satisfying assignment for such formulas
Finding Pseudorandom Colorings of Pseudorandom Graphs
We consider the problem of recovering a planted pseudorandom 3-coloring in expanding and low threshold-rank graphs. Alon and Kahale [SICOMP 1997] gave a spectral algorithm to recover the coloring for a random graph with a planted random 3-coloring. We show that their analysis can be adapted to work when coloring is pseudorandom i.e., all color classes are of equal size and the size of the intersection of the neighborhood of a random vertex with each color class has small
variance. We also extend our results to partial colorings and low threshold-rank graphs to show the following:
* For graphs on n vertices with threshold-rank r, for which there exists a 3-coloring that is eps-pseudorandom and properly colors the induced subgraph on (1-gamma)n vertices, we show how to recover the coloring for (1 - O(gamma + eps)) n vertices in time (rn)^{O(r)}.
* For expanding graphs on n vertices, which admit a pseudorandom 3-coloring properly coloring all the vertices, we show how to recover such a coloring in polynomial time.
Our results are obtained by combining the method of Alon and Kahale, with eigenspace enumeration methods used for solving constraint satisfaction problems on low threshold-rank graphs
The Lovasz number of random graphs
We study the Lovasz number theta along with two further SDP relaxations
theta1, theta1/2 of the independence number and the corresponding relaxations
of the chromatic number on random graphs G(n,p). We prove that these
relaxations are concentrated about their means Moreover, extending a result of
Juhasz, we compute the asymptotic value of the relaxations for essentially the
entire range of edge probabilities p. As an application, we give an improved
algorithm for approximating the independence number in polynomial expected
time, thereby extending a result of Krivelevich and Vu. We also improve on the
analysis of an algorithm of Krivelevich for deciding whether G(n,p) is
k-colorable
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